(TCO A) Consider the following age data, which is the result of selecting a random sample of 20 United Airlines pilots.
47 45 45 52 58 55 58 44 42 38
45 52 48 47 51 45 52 42 37 40
a. Compute the mean, median, mode, and standard deviation, Q1, Q3, Min, Max for the above sample data on age of pilots.
b. In the context of this situation, interpret the Median, Q1, and Q3.
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
For median, Q1 and Q3, arrange scores from lowest to highest. Median = 50th percentile, Q1 = 25th and Q3 = 75th. You should also be able to define min and max once you have done your arrangement.
121212121212
To compute the mean, median, mode, and standard deviation, as well as Q1, Q3, Min, and Max for the given age data, follow the steps below:
a. Computing the Mean, Median, Mode, and Standard Deviation:
1. Begin by arranging the data in increasing order:
37, 38, 40, 42, 42, 44, 45, 45, 45, 45, 47, 47, 48, 51, 52, 52, 52, 55, 58, 58
2. Mean: To calculate the mean, add up all the values in the data set and divide by the number of values (20):
(37 + 38 + 40 + 42 + 42 + 44 + 45 + 45 + 45 + 45 + 47 + 47 + 48 + 51 + 52 + 52 + 52 + 55 + 58 + 58) / 20 = 48.6
3. Median: The median is the middle value when the data is arranged in ascending order. Since we have 20 data points, the median will be the average of the two middle values:
Median = (47 + 47) / 2 = 47
4. Mode: The mode is the value that appears most frequently in the data set. In this case, the value 45 appears most frequently, so the mode is 45.
5. Standard Deviation: The standard deviation measures the spread or variability of the data set. There are multiple formulas to calculate it, but one common formula is as follows (using the sample standard deviation):
- Calculate the deviation of each data point from the mean: (37 - 48.6), (38 - 48.6), ..., (58 - 48.6).
- Square each deviation: (deviation1)^2, (deviation2)^2, ..., (deviation20)^2.
- Sum all the squared deviations: (deviation1)^2 + (deviation2)^2 + ... + (deviation20)^2.
- Divide the sum by the number of data points minus 1 (as it is a sample): Sum / (20 - 1).
- Take the square root of the result to obtain the standard deviation.
b. Interpreting the Median, Q1, and Q3:
In the given context, the Median represents the middle value of the ages. It indicates that half of the pilots are younger than 47 years, and half are older than 47 years.
Q1 (First Quartile) is the median of the lower half of the data. It represents the age at which the first 25% of the pilots fall below. In this case, Q1 is 45, indicating that 25% of the pilots are below the age of 45.
Q3 (Third Quartile) is the median of the upper half of the data. It represents the age at which the first 75% of the pilots fall below. In this case, Q3 is 52, indicating that 75% of the pilots are below the age of 52.